# SOS-SDP: an Exact Solver for Minimum Sum-of-Squares Clustering

arXiv: https://arxiv.org/abs/2104.11542 · HTML: https://mlsanity.com/paper/2104.11542/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2021.

## Built on by (most cited first, 14 of 14)

- [An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering](https://mlsanity.com/paper/2111.15571/) (2021)
- [An algorithm for clustering with confidence-based must-link and cannot-link constraints](https://mlsanity.com/paper/2212.14437/) (2022)
- [How to Use K-means for Big Data Clustering?](https://mlsanity.com/paper/2204.07485/) (2022)
- [Global Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming](https://mlsanity.com/paper/2209.08901/) (2022)
- [A Semidefinite Programming-Based Branch-and-Cut Algorithm for Biclustering](https://mlsanity.com/paper/2403.11351/) (2024)
- [Mixed-Integer Nonlinear Programming for State-based Non-Intrusive Load Monitoring](https://mlsanity.com/paper/2106.09158/) (2021)
- [Polynomial Optimization: Enhancing RLT relaxations with Conic Constraints](https://mlsanity.com/paper/2208.05608/) (2022)
- [Sketch-and-solve approaches to k-means clustering by semidefinite programming](https://mlsanity.com/paper/2211.15744/) (2022)
- [Predicting municipalities in financial distress: a machine learning approach enhanced by domain expertise](https://mlsanity.com/paper/2302.05780/) (2023)
- [A column generation algorithm with dynamic constraint aggregation for minimum sum-of-squares clustering](https://mlsanity.com/paper/2410.06187/) (2024)
- [Strong bounds for large-scale Minimum Sum-of-Squares Clustering](https://mlsanity.com/paper/2502.08397/) (2025)
- [A Scalable Global Optimization Algorithm For Constrained Clustering](https://mlsanity.com/paper/2510.22519/) (2025)
- [Unsupervised Machine Learning to Classify the Confinement of Waves in Periodic Superstructures](https://mlsanity.com/paper/2304.11901/) (2023)
- [Fast and effective algorithms for fair clustering at scale](https://mlsanity.com/paper/2605.13759/) (2026)

## Abstract (arXiv)

The minimum sum-of-squares clustering problem (MSSC) consists of partitioning n observations into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure where valid inequalities are iteratively added to the Peng-Wei SDP relaxation. The upper bound is computed with the constrained version of k-means where the initial centroids are extracted from the solution of the SDP relaxation. In the branch-and-bound procedure, we incorporate instance-level must-link and cannot-link constraints to express knowledge about which data points should or should not be grouped together. We manage to reduce the size of the problem at each level preserving the structure of the SDP problem itself. The obtained results show that the approach allows to successfully solve for the first time real-world instances up to 4000 data points.